The cell cycle is a fundamental process in systems biology, and Ordinary Differential Equations (ODEs) constitute a valuable tool for describing cell cycle dynamics. A major challenge in this field is estimating unknown parameters in these equations from sparse biological data. Recently, a class of machine learning methods, namely Physics-Informed neural networks (PINNs), has been proposed for solving ODEs and the corresponding parameter estimation problem. PINNs incorporate physical laws into the loss function of a neural network, with these losses evaluated at a set of scattered spatio-temporal points (called residual points). In this work, we propose a new point-wise weighting method for improving the accuracy of loss evaluation at these residual points, termed Time Scale Gradient Aggregation Self-Attention (TGSA). TGSA assigns dynamic weights to residual points by leveraging time-scale relationships and gradient information between state variables, capturing the internal dependencies among residuals. Our experimental results demonstrate the superior performance of TGSA in parameter estimation for four distinct cell cycle models.

In biochemical modeling, some foundational systems can exhibit sudden and profound behavioral shifts, such as the cellular signaling pathway models, in which the physiological responses promptly react to environmental changes, resulting in steep changes in their dynamic model trajectories. These steep changes are one of the major challenges in biochemical modeling governed by nonlinear differential equations. One promising way to tackle this challenge is converting the input data from the time domain to the frequency domain through Fourier Neural Operators, which enhances the ability to analyze data periodicity and regularity. However, the effectiveness of these Fourier based methods diminishes in scenarios with complex abrupt switches. To address this limitation, an innovative Multiscale Attention Wavelet Neural Operator (MAWNO) method is proposed in this paper, which comprehensively combines the attention mechanism with the versatile wavelet transforms to effectively capture these abrupt switches. Specifically, the wavelet transform scrutinizes data across multiple scales to extract the characteristics of abrupt signals into wavelet coefficients, while the self-attention mechanism is adeptly introduced to enhance the wavelet coefficients in high-frequency signals that can better characterize the abrupt switches. Experimental results substantiate MAWNO’s supremacy in terms of accuracy on three classical biochemical models featuring periodic and steep trajectories. https://github.com/SUDERS/MAWNO.